On definability in some lattices of semigroup varieties
Mariusz Grech, Olga Sapir · Semigroup Forum · 2012
An identity of the form x 1⋯x n ≈x 1π x 2π ⋯x nπ where π is a non-trivial permutation on the set {1,…,n} is called a permutation identity. If u≈v is a permutation identity, then ℓ(u≈v) [respectively r(u≈v)] is the maximal length of the common prefix [suffix] of the words u and v. A variety that satisfies a permutation identity is called permutative. If $\mathcal{V}$ is a permutative variety, then $\ell=\ell(\mathcal{V})$ [respectively $r=r(\mathcal{V})$ ] is the least ℓ [respectively r] such that $\mathcal{V}$ satisfies a permutation identity τ with ℓ(τ)=ℓ [respectively r(τ)=r]. A variety that consists of nil-semigroups is called a nil-variety. If Σ is a set of identities, then $\operatorname {var}\varSigma$ denotes the variety of semigroups defined by Σ. If $\mathcal{V}$ is a variety, then $L (\mathcal{V})$ denotes the lattice of all subvarieties of $\mathcal{V}$ . For ℓ,r≥0 and n>1 let $\mathfrak{B}_{\ell,r,n}$ denote the set that consists of n! identities of the form $$t_1\cdots t_\ell x_1x_2 \cdots x_n z_{1}\cdots z_{r}\approx t_1\cdots t_\ell x_{1\pi}x_{2\pi} \cdots x_{n\pi}z_{1}\cdots z_{r}, $$ where π is a permutation on the set {1,…,n}. We prove that for each permutative nil-variety $\mathcal{V}$ and each $\ell\ge\ell(\mathcal{V})$ and $r\ge r(\mathcal{V})$ there exists n>1 such that $\mathcal{V}$ is definable by a first-order formula in $L(\operatorname{var}{\mathfrak{B}}_{l,r,n})$ if ℓ≠r or $\mathcal{V}$ is definable up to duality in $L(\operatorname{var}{\mathfrak{B}}_{\ell,r,n})$ if ℓ=r.